We examine interactions between atoms continuously and coherently driven between the ground state and a Rydberg state, producing "Rydberg-dressed atoms." Because of the large dipolar coupling between two Rydberg atoms, a small admixture of Rydberg character into a ground state can produce an atom with a dipole moment of a few debye, the appropriate size to observe interesting dipolar physics effects in cold atom systems. We have calculated the interaction energies for atoms that interact via the dipole-dipole interaction and find that because of blockade effects, the R dependent two-atom interaction terms are limited in size and can be R independent up until the dipolar energy is equal to the detuning. This produces R dependent interactions different from the expected 1/R-3 dipolar form that have no direct analogy in condensed-matter physics and could lead to interesting quantum phases in trapped Rydberg systems.