The state-dependent multiple access channel (MAC) is considered where the state sequences are known strictly causally to the encoders. First, a two-user MAC with two independent states each known strictly causally to one encoder is revisited, and a new achievable scheme inspired by the recently proposed noisy network coding is presented. This scheme is shown to achieve a rate region that is potentially larger than that provided by recent work for the same model. Next, capacity results are presented for a class of channels that include modulo-additive state-dependent MACs. It is shown that the proposed scheme can be easily extended to an arbitrary number of users. Finally, a similar scheme is proposed for a MAC with common state known strictly causally to all encoders. The corresponding achievable rate region is shown to reduce to the one given in the previous work as a special case for two users.