First, see if the email is coming from a 'general' email address or specifically a hotel manager or the revenue/accounting department.
Before you try to negotiate you first need to decide how much you really want this specific property (and the amenities it offers... such as a kitchenette since eating out for an extended period of time can get costly) vs. 'any' hotel in the area that may be less expensive. If there are other hotels in the area that would be acceptable (ie- Residence Inn, Staybridge Suites, etc) then you have more leverage... if this is really the best hotel for your needs then you don't have as much room to negotiate.
Looking at the Hyatt website they are giving you a 0% discount ... for the dates you mentioned their online rate is $179/nite as well. It doesn't look like their AAA discount applies for such an extended stay, but you can probably do better by stringing together consecutive reservations (21 days for the first reservation and then 3-7 day increments from there) and using a AAA discount (it's not a big discount but even $8/nite adds up after 30+ nites)
I wouldn't expect them to take it, but i would start at $140/nite and offer to pre-pay for the first 30 days (hotels love to get revenue on the books). I would have a conversation something along the lines of 'I checked online and $179 is the same rate i can get by booking thru your website, i am expecting a discount given my 30+ nite length of my stay. If you could do it for $140/nite i would pre-pay for the first 30 nites now."
If they say 'no', then give it back to them... "how much of a discount could you offer if i pre-pay for the first month now?"
This is the time you begin to look at other hotels in the area that may work (i see there is also a Cambria Suites that seems to get good reviews, as well as other properties)... repeat the process with them to get your best rate. Once you have gotten everyone's best price, you can then:
1) negotiate further with your first choice of hotel to see if they will match the rate (or at least come close to it, throw in a room upgrade, etc)
2) try to shave a few more dollars off of the best rate you have been offered
The above assumes that you're sure of your start date as well as your need to be there for at least 30 nites, if not, adjust your numbers/dates accordingly.
One thing that may be working against you here is that the Hyatt House is an extended stay type of hotel, so a 30+ nite stay may not be such an anomaly for them (vs. say a 'regular' Hyatt Regency, Hyatt Resort, Marriott, Hilton, etc). As well, i would imagine that many of their extended stay customers are corporations who are paying for accommodations for temporary contractors or employees visiting from other locations, and they are less sensitive to price.
The above is how i would play it.
Good luck and let us know how it goes.