The CBB

Lancúhea:

Ceru dˈúmaçe gú.
[ˈkɛrɯ ðuˈmadʒə ɡuː]
1sg-AGT-VOL dream-INST locate-PRES-1sg
I am dreaming (because I want to have a dream).

Ci g'úmaçe gú
[kɪ ɡuˈmadʒə ɡuː]
1sg-PAT dream-INST locate-PRES-1sg
I am dreaming (because it is natural to do so; it just happens).

Riyan

Thosha
/θʊʃa/
dream-1S.NPST.IND
I am dreaming

Leopardish V

Yin wannar.
[jɪŋ‿w̞ɒnːa]
1s dream

Ythnandosian

Syŋŋerin sa.
/səŋ.'ŋe.rin sa/
dream-IND.PROG.NPST 1SG.ABS
I am dreaming.

Yabushionese

我、夢を見め。
Wai, yume ọ mime.
/wai jɯme ɔ mime/
1 dream ACC see-IMPERF
I am dreaming.

(I made a flag icon!)

Silvish

Gzo songi.
[dʑɔ ˈsõːʑɪ]
1SG.NOM dream-1SG.PRS
I'm dreaming

Mira
Niruxtam
dream-CAPT-1s
I'm dreaming.

iben kulon
iben kulo-n
1 dream-IPFV
[ibɛŋ‿kʉlɔn]

Anto

Da sumsa-da i'mom.
[taː ˈsʊmsada ˈimɔm]
1SG dream-1SG IMPF

Orkish

Zhuts zhutsampab.
[ʐut͡s ˈʐutsampˌab]
CONT sleep_dream-1SG.ABS

Elfish

ओकु तीमज़ द्रुगमे.
Ēku tīmas drugame.
[ˈeːk ˈtiːmas ˈθruɣam]
1SG.NOM PROG.PRS>1SG dream.INF

I am dreaming.

Miirna

Naay vuunaranna
naay vuunar-anna
1s dream-ACTIVE.PTCPL
[naːʝ ˈvuːnɐrɐnːɐ]~[nɐ ˈvuːnɐrɐnːɐ]

Japoné語

私 oué j'ga 夢見tèrë.
Ouétachi oué j'ga youmémitèrë.
[weˈtæʃi we ʒgɔ jʊmeˈmɪˈtæɛ̯ʁœ̈]
1SG TOP 1SG SUBJ dream-PROGPRES
I am dreaming.

In NoseBark:

Sya-r-zu vuz-lu
[sjar.zu vuz.lu]
sya=r=zu vuz=l-u
see=3=1 dream=INDEF-F
I'm dreaming.

Swahili:

Ninaota ndoto.
ni-na-ot-a ndoto
1s-PRES-dream-Ö dream(CL9)

Xingish

Mi drom.
[miː dro̞m]
1s dream
"I dream", "I'm dreaming", "I dreamt", etc.

The above expression can be interpreted in different tenses or aspects, depending on the context. To be precis, one could say, for instance:

Mi lig drom
[miː lig dro̞m]
1s lie dream
"I lie dreaming"

Undercommon

Zhedhņe antrem.
[ˈʐeɖ̚.ɳʲɛ ˈã.tɾɛm]
zhe-th-n ie an-trem
IPFV.TEL.NVOL-PHYS-1sg something make-dream
I am dreaming.

Iadhņe antrem.
[ˈjaɖ̚.ɳʲɛ ˈã.tɾɛm]
ia-th-n ie an-trem
IPFV.ATEL.NVOL-PHYS-1sg something make-dream
I dream/dreamed.

The two cases here differ only in whether the speaker has finished their dream or not.

Ular

我睡叫我。
Lăn thòm'áw lăn.
TOP.1SG dream 1SG
I am dreaming.

Pod
Óråvbos’yn atod
[ɤɹɑvpʼɑsin atɑd]
dream-IMP 1ST.NOM
I am dreaming

New Unnamed Project

Othi u reibeu.
['ɔ.tʰɪ 'u 're.ɓɯ]
PRS-see 1p.SG dream

Nothin' too wild.

Ganur Lon

Kup li'pop.
/kup liˈpop/
1SG-be COM=dream
I am with a dream. or I am having a dream.

Emilian (Dialàt Vgnulěṡ)
A soun a drě ch'a m'insógn.
/ɐ s̠ɒ̃ʊ̃n‿ɐ dreː kɐ mɪn'sŏɲ/
a soun a_drě_ca a m insógn
1SG be.1SG PROGR 1SG REFL.1SG dream.1SG
Emilian conveys the meaning of dreaming through a reflexive verb (insugnêres), quite curiously. Italian does not (it just uses "Sto sognando" as some of you already pointed out), and also, I have no idea where that in- prefix came from.

Häħadhvar/Hecathver
Vramiae evo.
/vra'mjae̯ 'εvo/
vramia-e evo
dream.1SG PROGR

Hajás
Jä ilbirüi.
/jε 'ilbiry.i/
jä ilbir-ui
PROGR dream-1SG
The <ui> ending changes to <üi> here, following the rules of vowel harmony. Curiously though, <ui> itself - used on its own as a 1SG subject pronoun - does not respect the rules of vowel harmony, as it features a back rounded vowel together with a front unrounded vowel. I figured I'd need exceptions to make the language more interesting...