|Districts moving left
Listed below are the districts that moved to the left by at least 3 PVI points (e.g. from D+11.9 to D+15.2). The first group contains 39 districts that are majority minority. There is good news for Republicans here. While Obama improving with the minority vote was a contributor towards his overall national vote margin, very few of these districts were in swing states and aren't districts the GOP will challenge in. There are a few notable exceptions, as two Republican held districts, FL-26 and CA-21 are on the list.
There were 14 districts that weren't majority minority that Obama also improved in. Fortunately, few were competitive, although the GOP does hold AK-AL, LA-1, and NY-11. Of those, however, only NY-11 could flip.
Districts moving right
Like the districts moving left, this isn't a varied list. In fact, 27 of the districts were in Illinois, Indiana, Missouri, Utah, and West Virginia. These were the five biggest improvements for Romney, outside of single district states. For the most part there's not a lot of excitement here, although there are a few swing districts. Democrats currently hold IL-10, IL-12, UT-2, and WV-3.
There were also 18 districts in other states. While the list above contains mostly conservative districts, this list contains districts the GOP might have been concerned about. GA-6 is suburban Atlanta. While GA-7 is a bigger concern, it's nice to see the Democratic vote hasn't spread here.
Several of the remaining districts, MI-1, ND-AL, SD-AL, VA-9, WI-7, and WI-8, swung to the GOP in 2010. In addition, there was concern about MT-AL and PA-6 flipping at some point. Unfortunately, there are no Democratic districts here.
Some districts that made big shifts 2004-2008 did shift again in 2012. There was a strong rightward shift in Massachusetts, Long Island, and Palm Beach County in 2008, but those districts didn't shift strongly again in 2012. There were a few that continued the trend:
KY-5: R+10 in 2004 to R+28 in 2012
WV-3: R+3 in 2004 to R+18 in 2012
WV-1: R+7 in 2004 to R+16 in 2012
VA-9: R+9 in 2004 to R+16 in 2012
MO-8: R+11 in 2004 to R+19 in 2012
If you look at the big shifts between the three elections you'll see that some parts of California are really moving away from the GOP. Some of them would've been competitive in 2004.
CA-46: R+1 in 2004 to D+11 in 2012
FL-9: R+0 in 1004 to D+11 in 2012
CA-41: R+0 in 1004 to D+11 in 2012
CA-21: R+6 in 2004 to D+4 in 2012
CA-51: D+8 in 2004 to D+19 in 2012
CA-35: D+8 in 2004 to D+17 in 2012
CA-29: D+19 in 2004 to D+27 in 2012
One phenomenon in California that was also exhibited nationwide was that Republican districts were more likely to get redder and Democratic districts were more likely to get bluer. While only 6 of 38 California Democratic districts got redder, 7 of 15 Republican districts did.