Based on a list compiled by Jon Nelson, last updated early September '98, but which now needs a new maintainer because Jon's busy looking after his newborn (congrats, Jon). Now there're thirty-six different ways; the target is still forty-two. Discussion on the newsgroup continues apace.

1. Don't panic (English)
2. Ne paniquez pas (or) Pas de panique (French)
3. Niente panico (or) Non lasciarsi prendere dal panico (Italian)
4. Nao se apavore (Portugese)
5. No te asustes (Spanish)
6. Tippanikjax (Maltese)
7. Psyhraimia (Greek)
8. Ná bíodh scaoll ort (Irish Gaelic)
9. Ny gow aggle (Manx Gaelic)
10. Ta det rolig (Norwegian)
11. Ta det lugnt (or) Ingen panik (Swedish)
12. Undgå panik (or) Slap af (or) Tag det roligt (Danish)
13. Geen paniek (or) Raak niet in paniek (Dutch - Netherlands)
14. Niet panikeren (Dutch - Flemish)
15. Moenie paniek nie (Afrikaans)
16. Keine Panik (German, thanks sauron)
17. Slappadu af (Icelandic)
18. Älä hätäile, or ei paniikkia (Finnish, thanks WWWWolf)
19. Ne panikui (or panikuy) (Russian)
20. Tea paanikasse sattuv (Estonian)
21. Nincs gáz (Hungarian)
22. Nedelat panika (or) Nedelat zdeseni (Czech)
23. Nie panikuj or Bez paniki (Polish, second corrected c/o Ariloulaleelay)
24. Al tilahetz (when addressing a male, or) Al tilahtzi (when addressing a female, or) Bli panika (if you're an American and don't mind sounding like one) (Hebrew)
25. Natarse (Farsi)
26. Ghabarao mat (Hindi/Urdu)
27. Ghabari beda (Kannada)
28. Awate nasanna (or) Awatenaideh (or) Shinpai suruna (or) Panikku wo okosuna (Japanese)
29. Uoki maka'u kuhewa (Hawaiian)
30. Tenang saja (or) Menenangkan (or) Tidak kelabakan (Indonesian)
31. Noli pavere (or) Nolite pavere (Latin)
32. Ne paniku (Esperanto)
33. Non affolla vos (Interlingua)
34. Ontday anicpay (Pig Latin)
35. Dubon't Pubanubic (Pig Greek)
36. yIlImQo' (Klingon)
37. No tinguis por (Catalan)
38. ko na xalni (lojban)
39. bu yao huang (Chinese)
40. mai' gyaa (Taiwanese, this and 39 c/o katana.)

(From the alt.fan.douglas-adams FAQ, with subsequent contributions / correction from Everythingites)