Rinse chicken and pat dry. Rub achiote paste all over chicken. In a bowl, mix tomatoes, lemon juice, ginger, garlic, thyme, salt, cumin, oregano, and pepper; add chicken and turn to coat. Cover with plastic wrap and chill at least 3 hours or up to 1 day.
Pour oil into a 10- to 12-inch frying pan over medium-high heat. When hot, add bell pepper and onion and stir often until limp, about 10 minutes.
Add chicken and its marinade to pan and bring to a boil; reduce heat, cover, and simmer until chicken is no longer pink at the bone (cut to test), about 10 minutes. With tongs, transfer chicken to a wide, shallow serving bowl; pour sauce over chicken.