October and November are the obvious months — clear skies, dry roads, post-monsoon greenery, and the country at its photogenic best. March to May is also reliable, with rhododendrons in flower on the upper trails to Singye Dzong.
The Lhuntse Tsechu in mid-December is worth timing for if you can handle cold weather; the dzong sits at a relatively low elevation but nights drop below freezing. The summer monsoon (June to September) brings landslides, leeches, and views obscured by cloud — avoid it unless you have a specific reason to be there.