Richard Grenville was the eldest son and heir of Sir Roger Grenville (d.1545), who was Captain of the Mary Rose when it sank in Portsmouth Harbour in 1545, by his wife Thomasine Cole, daughter of Thomas Cole of Slade. Thomasine survived her husband and remarried to Thomas Arundell. The ancient Grenville family were lords of the manors of Bideford in Devon and of Stowe, Kilkhampton in Cornwall. He was a cousin of Sir Walter Raleigh and the privateer Sir Francis Drake.