I aim more for an 'ein' sound. As in the german sense of phonics. But I picked your DINE-uh-stee as it is pretty close, in fact I am not sure if I could really differentiate the two now that I think about it.
This is one of those words in which there is a difference between American and British usage. The Oxford dictionary doesn't even give a long y as an alternative in 'dynasty' (only in dynast).