I was in Kenya beginning of June this year and didn't notice any poor weather. Just a sweet light sprinkling for 10 minutes in Samburu that turned the air to honey.

I was, however hoping for a bit of a thunderstorm in the Mara but only caught a threat of one far off in the distance.

Nights and mornings are a bit cooler so a light fleece and scarf if you're in an open vehicle would be recommended. T-shirts is all you need mid day when it warms up.

Nor did I notice any fewer wildlife just because it was "shoulder" season.

My safari in Samburu and the Masai Mara was private and I used the camps vehicles for my game drives. In both camps there were so few guests that I had a vehicle to myself the whole time.

The other advantage to this time of year is less vehicles at each sighting jockeying for the prime spot.

Here are a bunch of reports and comments on Kenya and TZ in May.

http://www.fodors.com/community/africa-the-middle-east/africa-blog-entry---trip-report-on-kenya-tanzania-safari-may-2008.cfm


http://www.fodors.com/community/africa-the-middle-east/trip-report-tanzania-kenya-zanzibar-with-micato-may-june-2006.cfm

http://www.fodors.com/community/africa-the-middle-east/ohmigosh-back-on-safari-with-micato---trip-report---may-2007.cfm


http://www.fodors.com/community/africa-the-middle-east/safari-pics-tanzania-may-2011.cfm

http://www.fodors.com/community/africa-the-middle-east/trip-report-may-2011-northern-circuit-tanzania.cfm


http://www.fodors.com/community/africa-the-middle-east/mara-trpi-report-aprilmay-2011.cfm

http://www.fodors.com/community/africa-the-middle-east/advice-needed-for-my-first-safari.cfm

http://www.fodors.com/community/africa-the-middle-east/tanzania-in-spring.cfm

I often travel in late-May/early-June (low season for rates in May, mid in June if in Kenya; though peak in June in Tanzania) or Nov thru mid-Dec (mid season rates, both countries) and never been stopped by rain.

Yes, an occasional shower or amazing thunderstorm, but these are brief and do not inhibit safari. Daytime temps are delightful, with cool mornings/nights (as it is year-round).

Prices are to your advantage, less visitors and so too less vehicles. And remember, the game does not go on holiday... they're there. In fact, during these past two years, some of the migration actually arrived early in the Masai Mara, though with over million animals there are numbers remaining in the Serengeti... the herds move on their own schedule.

Elsewhere in either country, also plenty of game. But if considering visiting at Tarangire (elephants, birds, baobab trees) in Tanzania, you'll find the grasses high, so better to stop here from late-June thru Oct. Alternatives for 'ellees' would then be Amboseli or Samburu in Kenya, with Samburu often preferred for the many unique species not found elsewhere.

In reality, anytime is good for safari.