Before looking at Reynolds's game, I want to mention that George Kelly hit a HR in three consecutive innings (4th, 5th, 6th) in a game in 1923 and that Mike Cameron had three HRs in the first three innings of a game in 2002, with two in the first inning and one in the third (he hit 4 HRs in the game).
Click here to see the box score of the Kelly game from Baseball Reference
Click here to see the box score & play-by-play of the Cameron game
Click here to see the box score & play-by-play of the Reynolds game
Here is what The Baseball Page says about his feat
"On July 2, 1930, in the second game of a doubleheader against New York in Yankee Stadium, Reynolds smashed three homers. Hundreds of players have hit three homers in one game, but the amazing thing about Reynolds' trifecta was that they came in the first three innings. Reynolds hit homers in each of the first three innings, including two of the inside -the-park variety. According to newspaper accounts, Reynolds lined two balls over the head of Yankees' center fielder Earl Combs and circled the bases before the relay throws made it back to the plate. His other homer was a blast off Red Ruffing into the left field seats. Reynolds' White Sox defeated the Yanks 15-4."I will put the link to that page at the end since the link was not working and I had to use the cached page. Another website says the non-inside the park HR was to RF. It is from the book '30: Major League Baseball's Year of the Batter by Ray Zardeito.
Two of the HRs were off of Hall of Famer Red Ruffing. Ruffing was a righty as was Reynolds, so he had no platoon advantage. Ruffing did not give up alot of HRs. In 1929, pitching for the Red Sox, he allowed 17 HRs in 244 IP. In 1930, it was 11 in 221 IP (with 1 in 24 IP with Boston) and in 1931 it was 11 in 237 IP. The other HR Reynolds hit was off of Ken Holloway, another righty.
Reynolds was 5 for 6 in the game, adding two singles later and he had 8 RBIs.
Here is the cached page from The Baseball Page