Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

Americans pronounce grout as GROWT (/graʊt/). You'll hear it in sentences like "He sealed the grout between the tiles to prevent water damage".
Record yourself saying "grout" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 4 sounds. Explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

Curl or bunch your tongue without letting the tip touch the roof of your mouth. Brace the sides of your tongue against your upper back teeth, and round your lips slightly.
Start with a dropped jaw and flat tongue. Glide into a relaxed, slightly rounded lip position as the back of the tongue stretches up.
Touch the tip or front edge of your tongue to the roof of your mouth just behind your teeth. Keep your jaw relaxed. Stop the air, then release with a puff.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "grout", the "t" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.