I was going to make a blog about this on the main site, but it feels like such a moment in time thing that I didn't see the point. For example, between doing this in January versus trying again in February, one of the soundtracks was removed (they used to have Ninja Scroll). So it feels like this might be better as a forum topic.
Commands are done with the fewest words to get the soundtrack to play. Any commands that are bold means it's available without Amazon Music, which is an additional fee. Also note that some commands work differently based on your service. Like "play Godzilla soundtrack" will play the 2014 Godzilla soundtrack if on Amazon Music, while it will play the 1998 GODZILLA album if just on Amazon Prime.
- Rodan (1956)
"play Rodan the movie soundtrack"
- Varan (1958)
"play Varan the Unbelievable soundtrack"
- Battle in Outer Space (1959)
"play Battle in Outer Space soundtrack"
- King Kong vs. Godzilla (1962)
"play King Kong vs. Godzilla original motion picture soundtrack"
- Destroy All Monsters (1968)
"play Destroy All Monsters original soundtrack"
- Akira (1988)
"play Akira soundtrack"
- My Neighbor Totoro (1988)
"play My Neighbor Totoro soundtrack"
- Porco Rosso (1992)
"play Porco Rosso soundtrack"
- Whisper of the Heart (1995)
"play Whisper of the Heart soundtrack"
- Princess Mononoke (1997)
"play Princess Mononoke soundtrack"
- GODZILLA (1998) {Album}
"play Godzilla album"
- Spirited Away (2001)
"play Spirited Away soundtrack"
- The Cat Returns (2002)
"play The Cat Returns soundtrack"
- Howl's Moving Castle (2004)
"play Howl's Moving Castle soundtrack"
- Tales from Earthsea (2006)
"play Tales from Earthsea soundtrack"
- Ponyo (2008)
"play Ponyo by Joe Hisashi"
- From Up on Poppy Hill (2011)
"play From Up on Poppy Hill soundtrack"
- Wolf Children (2012)
"play Wolf Children soundtrack"
- The Tale of the Princess of Kaguya (2013)
"play The Tale of the Princess of Kaguya soundtrack"
- The Wind Rises (2013)
"play The Wind Rises soundtrack"
- Godzilla (2014)
"play Godzilla soundtrack"
- When Marnie Was There (2014)
"play When Marnie Was There soundtrack"
- Shin Godzilla (2016)
"play Shin Godzilla soundtrack"
- The Red Turtle (2016)
"play Red Turtle soundtrack"
- Kong Skull Island (2017)
"play Kong Skull Island soundtrack"
- Pokémon Detective Pikachu (2019)
"play Pokémon Detective Pikachu soundtrack"
- Godzilla: King of the Monsters (2019)
"play Godzilla King of the Monsters soundtrack"
In addition to above, there is also the "Classic Themes from Godzilla" compilation. This is accessed by asking Alexa to play it, which has various themes from the Showa Godzilla films. Individual themes can be accessed from this compilation too, such as "play Godzilla vs. Megalon soundtrack" or "play Godzilla Raids Again soundtrack".
Remixes are also available by Alala and Monster Mania of Godzilla related themes. Each brought up by saying "play music by Alala" and "play music from Monster Mania".