We consider a generalization of the Gaussian dirty-paper problem to a multiple access setup. There are two additive interferences, one known to each transmitter but none to the receiver. The rates achievable using random binning schemes (i.e. schemes based on Costa's auxiliary random variables) vanish in the limit when the interferences are strong. In contrast, we show that lattice strategies ("lattice precoding") can achieve positive rates independent of the interferences. Furthermore, we derive an outer bound for the capacity region for arbitrary interferences, which is strictly smaller than the clean MAC capacity region. We then show that lattice strategies meet this outer bound for some combinations of noise variance and power constraints. In particular, lattice strategies are optimal in the limit of high SNR. Thus, the dirty MAC is another instance of a network setup, like the Korner-Marton modulo-two sum problem, where linear coding is better than random binning. We also derive lattice transmission schemes and conditions for optimality for the asymmetric case, where there is only one interference which is known to one of the users, and in particular for the helper problem, where the user which knows the interference does not have a message it wishes to transmit.