Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Americans pronounce countryside as KUHN-tree-sahyd (/ˈkʌntriˌsaɪd/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "He lives in a small village in the countryside".
Record yourself saying "countryside" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 9 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

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.

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.
Pull the corners of your lips back slightly. Arch the middle-front of your tongue high toward the roof of the mouth.

Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Start with your jaw open wide and your tongue resting low and flat. Glide the front of your tongue up toward the roof of your mouth as your jaw closes halfway.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "countryside", the "d" 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.
Stress falls on the first syllable, not the others. Stretch KUHN — keep everything else short and quick.