There are many languages that have true phonemic orthographies. An example is Wajarri (Pama-Nyungan family, Western Australia), which has a one-to-one correspondence between graphemes and phonemes.
As Wajarri has no voicing contrast but uses the "voiced" series to represent the plosives potential ambiguity between the velar nasal /ŋ/ and the sequence alveolar nasal + velar stop /nɡ/ is dealt with by placing a full stop between the two letters in the latter case i.e. ng vs. n.g. Many other Australian languages also have highly phonemic orthographies. Some other languages said to have phonemic orthographies are: Zambian languages and Austronesian languages (such as Malay and Maori).
Some European languages with very shallow orthographies are: Finnish, Croatian, Serbian. According to Seifart Croatian and Serbian are particularly shallow, even representing allomorphy according to the surface realisation, unlike Finnish which, while very shallow, does not show allomorphy.
Many of the languages with phonemic orthographies have comparatively little morphophonological variation and have had their orthographies developed relatively recently. As languages change, orthographies also must change or they will become gradually less phonemic. Languages with complex morphophonologies cannot be represented with a shallow orthography, though it's clear that English orthography is much more complex than it need be.