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In this paper, we present two simple Monte Carlo methods for estimating the achievable information rates of general finite state channels. Both methods require only the ability to simulate the channel with an a posteriori probability (APP) detector matched to the channel. The first method estimates the mutual information rate between the input random process and the output random process, provided that both processes are stationary and ergodic. When the inputs are iid equiprobable, this rate is known as the Symmetric Information Rate (SIR). The second method estimates the achievable information rate of an explicit coding system which interleaves m independent codes onto the channel and employs multistage decoding. For practical values of m, numerical results show that this system nearly achieves the SIR. Both methods are applied to the class of partial response channels commonly used in magnetic recording.