In Kenya and Tanzania, the highlight is the Great Migration, best from July to October when the Mara River crossing takes place. Game is present all year, however, but the long rainy season in March to May (with the heaviest rains from April to May) is not the best time to visit as wildlife disperses and is hidden by vegetation, and rain can make roads impassable. However, there are some great deals available during this period, so it's worth considering if you're on a really tight budget. Be aware that Kilimanjaro climbs are risky in the wet season, and that afternoon downpours are heavier along the coast and islands. Some lodges in remoter southern Tanzania will close during this period.
For gorilla trekking in Uganda or Rwanda, be prepared if you are going during the rainy seasons of November, and from March to May – especially in Uganda, where treks usually take longer. It doesn’t rain all day every day, but comes in big bursts, and gorilla tracking permits may be heavily discounted – but if you aren’t a fan of rain, think about booking outside the rainy season.