How can a cleric cast a spell meant for clerics of a deity other than his own?

Question

Some cleric spells have deity designators. For example the spell dread blast says

Level: Cleric 4 (Cyric)

Is there a way for a cleric to use this spell without worshiping Cyric?

Answer 1

Any cleric can cast such spells, but beware of updated spells

What it means when a spell has a deity's name after the spell's name is explained in Magic of Faerûn in the chapter Spells on Origin:

Some spells have an origin given in parentheses after the spell's level. This indicates the deity that initially provided this spell (such as Azuth), a group of spellcasters that created the spell (such as the Red Wizards), or a region where the spell originated (such as Halruaa). Since these spells have been created, others have duplicated the efforts of the creators, so that now the cleric spells created by Azuth are available to all faiths, spells made by the Red Wizards have made it into other hands, and Halruaan spells have spread beyond that remote country. (86)

(Emphasis mine.) So, while the 4th-level cleric spell dread blast [evoc] (Lords of Darkness 186) originated with clerics of Cyric, any cleric could cast it.

I say could because for the 3.5 revision the Player's Guide to Faerûn updated the spell dread blast (101-2) to a 4th-level spell exclusive to the feat Initiate of Cyric (80) and also updated several other spells that formerly had deity designators. Player's Guide to Faerûn on Deity-specific Spells has this to say about that:

Some of the deity-specific spells included in the initiate feats [that appear in the Player's Guide to Faerûn] previously appeared in Magic of Faerûn [and, apparently, other sources] with a deity designator [as explained above], but there was no rule prohibiting their use by other clerics. If you use initiate feats [that appear in several sources including Player's Guide to Faerûn] in your game, you should not allow clerics without the appropriate initiate feat to cast those spells. (79)

Hence, the spells that were not updated for the 3.5 revision that carry a deity designator remain available to all clerics, but the really restricted spells are those that appear only as part of an initiate feat—and that latter group now unfortunately includes the spell dread blast. (While it's woefully incomplete and, of course, out of date—and even lacks the spell dread blast—, a good place to start searching for what spells have been updated for the 3.5 revision is the Rules Reference column "Feats, Prestige Classes, and Spells.")

Answer 2

RAW you are able to learn any spell under a "Cleric Spells" list upon leveling. This comes from the way that Clerics learn new spells

Spells Gained at a New Level
Characters who can cast divine spells undertake a certain amount of study between adventures. Each time such a character receives a new level of divine spells, he or she learns new spells from that level automatically.
Independent Research
A divine spellcaster also can research a spell independently, much as an arcane spellcaster can. Only the creator of such a spell can prepare and cast it, unless he decides to share it with others.

So there is always a way for you to learn the spell. Weather you learn it through your own studies, or the creator of the spell can always teach it to you.

From what I've seen in my experience with the Forgotten Realms setting that this spell comes from is that the churches each have their own signature miracles. (Which to my group made sense, each church would find its own way to channel divine magic in different ways) but it is still just learning a new method of channeling divine magic. It may be difficult to get enough information on another churches practices and magic to learn their spells without being taught it by one of their members, but it should be possible.