The short answer can be found in these maps from Professor Bert Vaux's Dialect Survey:
The speech accent archive, suggests that the -dee ending is popular in the American Southeast, particularly in Louisville, Kentucky; Atlanta, Georgia; Belmont, Mississippi; Plantersville, Arkansas; Elmore, Alabama; and Pensacola, Florida.
Additionally, I invite you to google mondee tuesdee or mondy tuesdy, or similar. The eponyomous search engine will return many thousands of hits. Depending on how many and which of them you investigate, you are likely to learn one or more of the following things:
The Mundee-Tuesdee people are a hardy lot. While a large group of them may be oldsters and on the way out, they are alive and well in many regions of the good ol' US of A. You'll find them in Baltimore, Phuldulphia, Pittsburgh, Utah, Misouri, Fort Worth, Houston--in fact, all of The Drawling South. I can't find a link to prove it, but Midwesterner and radio personality Garrison Keillor is a Mundee-Tuesdee person. And, I wouldn't link to her if I could, but guess how Alaska's Sarah Palin pronounces the days of the week.
The Mundee-Tuesdee people are often disdained for a perceived lack of intelligence. (As a native of Ballmer, Merlin myself, I bear the shame of Mundee-Tuesdee heritage, though I have renounced it.)
It's difficult to find online academic sources addressing this topic by googling mondee tuesdee or similar pronunciation approximations.
Most online dictionary entries, including Cambridge Dictionaries Online and Dictionary.com give both pronunciations, with the -day ending listed first.
A post by expert at usingenglish.com offers this partially contrary but more detailed entry from Longman Pronunciation Dictionary:
Although Received Pronunciation and General American are both traditionally considered to prefer di, most speakers in practice use both pronunciations for this suffix, often in a strong form—weak form relationship. The deɪ form is generally preferred in exposed positions, for example at the end of a sentence: I’ll do it on Monday ˈmʌn deɪ ; the di form is preferred in close-knit expressions such as Monday morning ˌmʌnd i ˈmɔːn ɪŋ ǁ -ˈmɔːrn-
Extensive investigation of the Google results for the related IPA symbols may yield a precise answer to the OP's query: What US dialects characteristically use this pronunciation, however, a nearly identical question is asked and not definitively answered at the Linguistics Stack Exchange site. Excerpts include:
There's lots of anecdotal information on this, like this English.Stackexchange question, which is interesting enough, but I'm curious if anyone knows of scholarly work on the subject. -Dec 24 '11 at 5:08 by Mark Beadles
I have found one scholarly work on the subject, K. Wheatley in American Speech, Vol 9 No 1, Feb 1934, pp 36-45, "Southern Standards". Author writes:
Yesterday, Monday, Tuesday, etc., always have [i] in the final syllable in Southern speech while [ei] is often heard in these words in the linguistic West.
It is not clear what dialects she means by "Southern" and "Western" other than that these are American dialects.
Mark Beadles, the author of the excerpts has provided this information in his answer here as well.
As I know from personal experience, and as I've demonstrated with the evidence (albeit anecdotal) at the links above, the distribution of these pronunciations is far more complicated than this Southern vs Western dialectical dichotomy can easily account for. I can vouch for a strong preference for the -dee ending in a certain group of native Philadelphia area and Baltimore area English speakers.
I also believe that the following linguistics.stackexchange comment, especially regarding the influence of socioeconomic factors, is accurate.
@MarkBeadles -- It may well not be a dialectal matter. Individuals vary a lot in how their final vowels get reduced in rapid speech, even from hour to hour, or mood to mood. Not to mention that socioeconomic factors like class, income, education, race, and status are more often correlated than geographical location. Except of course where there's cross-correlations already. – jlawler Dec 24 '11 at 19:27