As one of the three major Scandinavian languages, Danish is closely related to Swedish and Norwegian but merits special treatment due to a few important differences in pronunciation. It shares with these languages an almost pathological lack of direct correspondence between letters and sounds, so do the best you can. If it's any comfort, English is probably even harder to pronounce, for those who don't speak it.
Danish uses ä, æ, å and ø the same as in Norwegian and Swedish.
|ä, æ||a, as in "ash", but the eh approximation of German will do|
|å, aa||aw or oh|
|e (final)||uh, like German unstressed final e|
|i||ee, or shortened to ih (before a double consonant)|
|ø||ö, like German or Swedish ö, or French eu|
|u||u, as in "put"|
|y||ü, like German ü, or French u|
These are complicated because they usually have ambiguous spellings: combinations such as av and og may be pronounced as they look, or they may be diphthongs (the consonant is swallowed while the vowel sound is changed). In general, Danish forms diphthongs by combining vowels with the letters v and g: then the sound of v becomes w and the sound of g essentially becomes y. It should be emphasized that the following suggestions are approximate.
|av, ag, af||ow, e.g. Vagn Holmboe = vown holm-boh|
|ej, eg||iy, like the word "eye" (this is analogous to German ei)|
|ey||ay, do not confuse with ej and eg|
|øv||öw, something like the British pronunciation of "oh"|
|ov, og||ow or aw, e.g. Mogens Wöldike = mo-wens völ-di-keh|
|yv||üw, do the best you can, or pretend you're Scottish and say "ew"|
Keep in mind that these combinations may not always be intended as diphthongs; in particular, if one of these is followed by another vowel, it's probably best to pronounce it just as it looks. However, if one of these combinations is followed by a consonant to create an uncomfortable looking consonant cluster, the diphthong pronunciation may be a good solution to the problem.
|d (after a vowel)||th, voiced as in English "the", e.g. Niels Gade = neelz gah-thuh|
|g (between vowels)||silent (often, but not always)|
|g (otherwise)||g, hard as in "get", does not soften as in Swedish and Norwegian|
|k||k, always hard as in "keep" (not like Swedish/Norwegian)|
|nd (final)||n, e.g. Sangkor Sokkelund = sahng-kohr soh-kuh-lun|
|r||r: uvular (gutteral) trill, something like French or German, though not altogether dissimilar from English|
|rd (final)||r, e.g. Per Nørgård = pehr nör-gohr|
|s||s (NEVER z like German or Italian)|
|st, sp||st, sp (NOT like German sht, shp)|
As with Swedish and Norwegian, the German rules for stress apply similarly to Danish. Stress falls normally on the first syllable, unless this syllable is one of a certain set of prefixes. In names, the first syllable is almost always stressed.