The Romans did, in fact, adopt many things from other civilisations such as the stereotypical style of helmet (from Gaul) and there was much religious syncretism in religious affairs, such as the adoption of Mithras and Epona. However, this was not to the same extent as Greek culture.

The reason why the Romans adopted a lot of Greek culture and architecture is because Greek culture simply was the most high-brow and geographical proximity. The Greeks had philosophy, drama, history, impressive buildings and a very nice sounding language. Furthermore, thanks to Alexander and earlier Greek colonists, Hellenic influence had spread across the Mediterranean with even the Carthaginians adopting certain elements of Greek culture. This was something that the Celts and Germans did not have (barring the language), and they were too far away from India and China to adopt any of their customs. To assume that the adoption of Hellenic culture was fast, however, is not entirely true. Hellenisation of the Roman world took a significant amount of time. The time period between the use of phalanxes in the early Roman military to the adoption of the Aeneid (which, amongst my Classical Civilisation class, is sometimes labeled the best fanfiction of the ancient world) is a very long period of several hundred years.

Furthermore, the Romans did not adopt everything about Greek culture. The east was often portrayed as a land of decadence, femininity and luxury in comparison to the more masculine Rome. Cato the Elder was a famous politician who opposed the adoption of Hellenic culture as well. There are also several differences between the democracy of Athens and the republic of Rome and Rome had a much deeper class system than Greek democratic states (not to say that the Greeks did not). Woman in Rome were generally treated a lot better than in Athens, for example the use of the veil was not a common feature of Roman society. Judging the different portrayal of slaves in Aristophanes', say, 'Frogs' and Plautus' 'Pseudolus' or 'Miles Gloriosus', slaves probably had a better position in society than in Greece. Interestingly, it appears that no Roman Emperor from antiquity was ever born in Macedonia or mainland Greece apart from emperors born in Constantinople.

So in conclusion, the reason why a lot of Greek culture and architecture was adopted by the Romans was because it was perceived to be the greatest culture in the Mediterranean. The process of Hellenisation took hundreds of years and never was complete. There were also other external influences but no single influence was as significant as the Greek influence. Hoped this answered your question.