Well, word order is a pretty difficult subject that only recently has gotten a fair amount of theoretical attention. Previously, what word order conveys had to be internalized by scholars and exegetes through reading a lot of Greek (i.e., more than what's in the NT).
Generally, Greek uses word order to convey how parts of the sentence relate to the larger context. (Technical terms for this include "information structure," "focus," and "topic"; less technical is the term "emphasis"). Since most grammars and teaching materials deal pretty much only with the meaning of the sentence, they tend to overlook these discourse-related effects.
The default order with the genitive after the noun conveys nothing special. This pattern is in the example that Funk quoted.
If the genitive comes between the article and the noun, then it tends to indicate that the genitive is more prominent than the noun, e.g., Rom 3:25 ὃν προέθετο
ὁ θεὸς ἱλαστήριον διὰ πίστεως ἐν τῷ αὐτοῦ αἵματι. This pattern is fairly rare.
If the genitive comes before the noun, what it conveys may depend on the position of other words in the sentence. If the genitive is first in the clause, it may be that the genitive is prominent, for example, Eph 2:10 αὐτοῦ γάρ ἐσμεν ποίημα. Occasionally, an initial genitive is merely contextualizing, for example, Matt 1:18 Τοῦ δὲ Ἰησοῦ Χριστοῦ ἡ γένεσις οὕτως ἦν. On the other hand, if the genitive is not first and follows another constituent (usually the verb), then it tends to be the case that the constituent it follows is prominent, e.g., John 19:29 σπόγγον οὖν μεστὸν τοῦ ὄξους ὑσσώπῳ περιθέντες προσήνεγκαν αὐτοῦ τῷ στόματι. This last pattern is perhaps the most common variant order.
I fear that even this explanation has oversimplified the complexities of Greek word order. My advice to beginners is to concentrate on understanding what the sentence means first and then read a lot of Greek in context to begin to internalize how the word order relates to how the discourse is structured.
Stephen C. Carlson, Ph.D.