I would be interested in which game you can give for an example, where attacked so heavy? Was it blitz type game or standard. You can just say which game in your "my games". I have found that when one comes out shooting and not a GM player, you can develop your pieces very fast, and when there attack fizzles out they are in a world of hurts position wise, and you are now ready to launch a supported attack!
How to Counter Relentless Attackers

I would be interested in which game you can give for an example, where attacked so heavy? Was it blitz type game or standard. You can just say which game in your "my games". I have found that when one comes out shooting and not a GM player, you can develop your pieces very fast, and when there attack fizzles out they are in a world of hurts position wise, and you are now ready to launch a supported attack!
Yeah, this is for sure the way to handle situations like that.
When you encounter an 'opening' you don't know, just follow some basic opening principles:
Get your pieces developed as efficiently as possible.
Try to control the center.
Don't move a piece twice unless it is threatened or can gain some solid advantage.
Take care of your king safety.
Get your rooks connected.
When you are being attacked during the opening, try to make moves that help accomplish one of these goals. Don't get sucked into playing their game.
Once you are developed, look for solid opportunities to get an advantage. When someone comes out fighting, they often neglect one of the following things:
Leaves pieces or pawns under protected - Find a way to overpower the under protected piece.
Gets far behind in development - When you have all or most of your pieces developed and your opponent has only developed 2 or 3 pieces with the intent to try and attack, then you can begin using your mobile force to constrict your opponents undeveloped pieces.
Leaves the king uncastled - This is a big-ass red flag when you are developed and your opponent has not yet castled. Find a way to prevent the casteling or capture the king. A situation like this can be ripe with opporunites to make sacrafices to force a win.

Attacked with the Morra? Well as long as you don't fall into any traps you should be fine. It's just a bit uncomfortable to find a good place for your queen so I usually put it on b8.

But...but...I ususally come out swinging! There's no fun in just setting up for the first 10 or so moves, staring at your opponent across the board slowly playing some elaborate, memorized opening. In my opinion if your not playing at a super high level, then they wont be able to react far from their memorized lines to whatever you throw at them. Anybody ever read Ender's Game?

How can I try to counter a relentless, accurate attacker?
The remedy against wild men and crazed attackers is suffering tight,tight chess don't give him any air to breath then he won't be able to charge up the board and launch those stormy attacks like a hockey player on a breakaway.
Shut him down and don't let him do his thing!

One time I was steamrolled by an "relentless attacker" type, but what I thought was mindless aggression was actually a big attack I walked into because I didn't know the opening. Basically I failed to create any counterplay so my opponent had a free hand in his attack.
So the opening is something to consider.
But creating counterplay is the most general answer. Passive defense is both technically and psychologically difficult. Open lines somewhere, anywhere else, and create some kind of threat. Ideally you open lines in the center. This is all the more effective if they're hyper-aggressive and attack before finishing development.

... There's no fun in just setting up for the first 10 or so moves, staring at your opponent across the board slowly playing some elaborate, memorized opening...
I love owning territory. I don't memorize openings, but I try to develop and restrict as much as my imagination will let me.
Own the board until the other guy can't move and break through as soon as there is an opening. I find strategic planning much more satisfying that tactics to be honest.
Playing at such a low level as I am, that foggy 10 move break-through plan that actually comes together is much more satisfying than spending every game trading pieces and pawns from move 3 onwards until the game ends. Blergh!!! What a boring one-sided way to play chess :)

But...but...I ususally come out swinging! There's no fun in just setting up for the first 10 or so moves, staring at your opponent across the board slowly playing some elaborate, memorized opening. In my opinion if your not playing at a super high level, then they wont be able to react far from their memorized lines to whatever you throw at them. Anybody ever read Ender's Game?
No, but playing 10 moves in a few minutes to reach a dynamic unbalanced position whose general ideas are familiar to both you and your opponent is exciting and makes for an interesting game.

That is a good question and I read very nice answers. Kleelof's answer was very nice and I saved parts of it although I find the pic bothersome. SOme of the guys about following opening principles and "just open lines" and stuff like that, well, Smith-Morra Gambit is a bitch created to answer that kinda aloof approach. I encountered one and it is a terrible opening for players of my level. Good luck when you face one against somebody who knows what they are doing.
Matt, my approach is I just read everything. I have found that I feel comfortable with Chicago defense against Smith-Morra. There is a small pamphlet on Wikipedia you can download for free. I just got this book by Richard Palliser-Fighting the Anti-Sicilians_ Combating 2 c3, the Closed, the Morra Gambit and other tricky ideas (Everyman Chess 2007)
It's almost never played at elite level. So I guess the path is straight forward... :)

Attack him back offence is the best defence
I thought a good defense is the best offense.

If your opponent is really a relentless and accurate attacker, you're in real trouble! Luckily, there aren't many of those around.
I based my style against that kind of player. I need to be attacked aggressively for me to feel comfortable. I cannot stand playing a turtle/hippo type of player. I seem to falter against those.
I played a game at a tournament recently against someone who I had just played in my last tournament. I lost both times, even though they are a bit lower rated. I think this can happen multiple times because some types of players just play well against other types of players. This isn't the first opponent that has had my number, or vice versa.
How can I try to counter a relentless, accurate attacker? They always play gambits or openings that in general are out of the ordinary and allow attacks. (Smith-Morra Gambit in the Sicilian, for example) I want to learn how to turn this around.
I can post a game if neccisary. Thanks.