MCMC updates of Dirichlet distributed variables

Say Theta = (theta_1,...,theta_K) are some model parameters representing a discrete distribution:

for all k theta_k >=0, and sum theta_k = 1

I have some data X. I know how to compute P(X|Theta) and P(Theta).

I want to sample P(Theta|X) using MCMC moves. I start with some random initialization for Theta, and in each step I sample a new Theta':

T(Theta --> Theta') ~ Dirichlet(mu*Theta)

where mu is the pseudo sample size (constant).

I accept with probability:

P(Theta') P(X|Theta') Dir(Theta; mu*Theta') / [P(Theta) P(X|Theta) Dir(Theta'; mu*Theta)]

For some reason this doesn't work (likelihood rapidly drops even when I initialize with correct state).
1) Is there a chance I am missing a Jacobian term?
2) Is there a more standard proposal distribution to use in this case (other than sampling from the prior of course).