Run cold water over the rice in a strainer for 1 minute, lifting rice to rinse thoroughly. In a saucepan, combine rice and the 2-1/4 cups water. Bring to boiling; reduce heat. Cover; simmer for 40 to 45 minutes or until done. Drain off any excess liquid. Fluff with a fork; cool until rice is just warm.
In a large mixing bowl, combine whole-wheat flour, oats, rye flour, and yeast. Stir in the 1-1/4 cups warm water, the honey, and salt. Stir in rice. Cover; let stand at room temperature for 4 to 6 hours. (Refrigerate overnight, if you like.)
Using a wooden spoon, stir in as much of the bread flour as you can.
Turn dough out onto a lightly floured surface. Knead in enough of the remaining bread flour to make a moderately stiff dough that's smooth and elastic (6 to 8 minutes total). Shape into a ball. Place in a lightly greased bowl; turn once to grease surface. Cover and let rise in a warm place until doubled (1 to 1-1/2 hours).
Punch dough down. Turn out onto a lightly floured surface. Divide in half. Cover; let rest for 10 minutes. Shape each portion into a round loaf. Place on a greased baking sheet and flatten slightly to 6 inches in diameter. (If you prefer a rounder loaf, flatten dough to 5 inches in diameter.) Cover; let rise until nearly doubled (40 to 50 minutes). If you like, sprinkle dough with additional oats.
Bake in a 375 degree F oven for 40 to 45 minutes or until bread sounds hollow when tapped. Cover loosely with foil the last 10 minutes, if necessary, to prevent overbrowning. Brush lightly with the cooking oil. Remove; cool. Makes 2 loaves (24 servings).