Greyhound tracks keep records of all of their races and most will provide results to anyone who contacts them. Most tracks have websites with contact information. Either use a search engine to find the website or visit Greyhounds.com, which has links to many greyhound track websites. In many cases the track websites also have the results of races available. For example, the Birmingham Race Course has a page of results for the most recent two months of races.
Another way to find race results is to visit Trackinfo.com. On the website, click Greyhounds at the top of the page, then click Results. This page includes a list of all of the greyhound tracks; find the desired track and click the Results link for it. On this page, select the desired date and race number to see the results from that race. In addition to the placement information, this also shows the payouts for different winning bets, such as getting the trifecta. The information on Trackinfo.com is comprehensive and dates back several years. For example, there are results for the Birmingham Race Course for each day back to the beginning of 2010.
Any organization that accepts bets on greyhound races should have results for the races it covers available upon request.