This question depends a lot on what you mean by “that time period.” Sparta was around for a long time, lasting from the bronze age until its sack in 396 AD. If we are just talking about the classical period of Greek history, the 5th and 4th centuries BC, then the answer depends on how you define Athenian. Later Roman authors wrote extensively about Sparta, both about Sparta in their own day, and histories of classical Sparta. However, when Roman authors wrote about classical Sparta, they were still relying heavily on Athenian sources. Poetry is the best source we have that gives us words written by native Spartans, although extent Spartan poetry comes mostly from the 7th and 6th centuries BC, long before the classical period.

Starting with are usually the best sources, authors of prose history in classical Greece, there are several major authors who might be considered non-Athenian, but all of them still have significant ties to the city. Herodotus is one of the most important sources on 5th century BC Greece, including Sparta, and he was not Athenian, at least not by birth. However, he did spend a great deal of time in Athens, and clearly wrote a significant amount after he moved to Athens. Other ancient sources tell us he was buried in Athens, so he hardly counts as a non-Athenian source. The other major surviving works of history that were written 5th and 4th century Greece were mostly by Athenians as well, although Xenophon poses an interesting case. Xenophon is one of our best sources on Sparta specifically, due to his first hand knowledge of the Spartans. He was born to a wealthy Athenian family, but left the city at a young age as a mercenary and he did not spend significant amounts of his adult life in Athens. At times, he fought alongside the Spartan army and became friends with a Spartan king. He spent at least 20 years of his life living in Sparta, so he can be counted as at least partially Spartan. However, he still is an Athenian by birth, and he wrote in Attic Greek, the dialect of Athens, not Doric, the dialect of Sparta. All of the other major historical works that deal with Sparta from the classical period that have meaningful extant text from the classical period are from solidly Athenian authors. A possible exception to this is Aristotle, since he was not a native born Athenian, but he composed his writings about Sparta while living in Athens and they reflect Athenian influence at least somewhat, so he doesn’t really count either.

Later Roman historians spent a great deal of time writing about Sparta. Many Romans were fascinated by Sparta, and we have a lot written by Romans about classical Sparta that is of questionable quality. However, many of these Roman sources are still very valuable. Authors such as Plutarch and Diodorus Siculus wrote about Sparta based on many sources that no longer exist, and they are a window into these now lost or partially-lost sources from classical Greece. For example, Plutarch presents a collection of “Sayings of Spartan Women,” which are only extant in Plutarch’s work. Plutarch was writing centuries after these sayings would have been said, so the authenticity is difficult to establish, but nonetheless they are a fascinating and unique insight into women in Sparta. However, even the Roman authors run back into the problem of Athenian dominance of the written record. Even though Plutarch and other Romans had access to far more sources then we do today, most on the topic of Sparta were still written by earlier Athenian authors. Plutarch admits in biographies of the 3rd century BC kings of Sparta Agis IV and Cleomenes III that he relied heavily on the contemporary writings of Phylarchus of Athens.

Although I suspect you are more interested in classical Greece, we can learn a lot about Roman era Sparta from a number of different contemporary Roman authors. Cicero tells us that many elite Romans travelled to Sparta as tourists to watch the ancient religious festivals of the city. The third century AD historian Herodian wrote that in 217 AD, the emperor Caracalla recruited 500 Spartans to his army to form a “phalanx.” This may have been an attempt to play on the reputation of the classical-era Spartans, and it certainly reflects the Roman fascination with Sparta.

Since you specified “good” sources, I first went over ancient historical texts, which were intended to record history. However, when faced with such sparse material and significant bias in the origin of the sources, one begins to turn to other options. Poetry written by Spartans is an extremely valuable resource, especially since it is through poetry that we finally can read the words of native Spartan authors. Spartan poetry from the classical period is fairly sparse, which may reflect a lack of poetry being written by Spartans at this time. However, we do know that Spartan armies in the classical period heard martial poetry read aloud, and while it is possible that they were only reading centuries-old poetry, it is more likely that at least some new compositions were made during this time, and that these did not survive. Regardless of why so little poetry survives from classical Sparta, our best examples of Spartan poetry date to earlier centuries, from the so called “archaic period.” All of these authors survive only in fragments, preserved in quotations by later authors and Hellenistic-era papyri excavated from Egypt. These poets wrote in Doric Greek, the dialect of the Spartans, or an Ionic poetic dialect based on Homer, unlike all of the prose historians, who wrote in Attic Greek.