I think it depends on your preferred Black opening. If you play the King's Indian, for example, you can get to similar positions by playing the Pirc or Modern Defense. If you play the Nimzo-Indian or some such, though, the answer is, not really.
The problem with doing so is that all the most common replies to 1.d4 (d5, Nf6, even f5) focus on immediately preventing e4. If White plays e4 right off the bat, your strategy is bust before you even get to make a move.
This is really the essence of why I feel 1.e4 is the strongest White opening -- you take advantage of your one opportunity to play it unhindered.
I have found that I play far better against d4 than against e4 when I play as black. Does anybody know of a reliable opening that can transpose to a closed d4-style position? Thank you.